Solve (4/3)^x=(27/64)

$$( \frac { 4 } { 3 } ) ^ { x } = ( \frac { 27 } { 64 } )$$

Solve for x

$$x=-3$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$\left(\frac{4}{3}\right)^{x}=\frac{27}{64}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{4}{3}\right)^{x})=\log(\frac{27}{64})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$x\log(\frac{4}{3})=\log(\frac{27}{64})$$

Divide both sides by \log(\frac{4}{3}).

$$x=\frac{\log(\frac{27}{64})}{\log(\frac{4}{3})}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$x=\log_{\frac{4}{3}}\left(\frac{27}{64}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{4}{3})}+\log_{\frac{4}{3}}\left(\frac{27}{64}\right)$$

$$n_{1}\in \mathrm{Z}$$